# Documentation

Mathlib.LinearAlgebra.ExteriorAlgebra.Basic

# Exterior Algebras #

We construct the exterior algebra of a module M over a commutative semiring R.

## Notation #

The exterior algebra of the R-module M is denoted as ExteriorAlgebra R M. It is endowed with the structure of an R-algebra.

Given a linear morphism f : M → A from a module M to another R-algebra A, such that cond : ∀ m : M, f m * f m = 0, there is a (unique) lift of f to an R-algebra morphism, which is denoted ExteriorAlgebra.lift R f cond.

The canonical linear map M → ExteriorAlgebra R M is denoted ExteriorAlgebra.ι R.

## Theorems #

The main theorems proved ensure that ExteriorAlgebra R M satisfies the universal property of the exterior algebra.

1. ι_comp_lift is the fact that the composition of ι R with lift R f cond agrees with f.
2. lift_unique ensures the uniqueness of lift R f cond with respect to 1.

## Definitions #

• ιMulti is the AlternatingMap corresponding to the wedge product of ι R m terms.

## Implementation details #

The exterior algebra of M is constructed as simply CliffordAlgebra (0 : QuadraticForm R M), as this avoids us having to duplicate API.

@[reducible]
def ExteriorAlgebra (R : Type u1) [] (M : Type u2) [] [Module R M] :
Type (max u2 u1)

The exterior algebra of an R-module M.

Instances For
@[reducible]
def ExteriorAlgebra.ι (R : Type u1) [] {M : Type u2} [] [Module R M] :

The canonical linear map M →ₗ[R] ExteriorAlgebra R M.

Instances For
theorem ExteriorAlgebra.ι_sq_zero {R : Type u1} [] {M : Type u2} [] [Module R M] (m : M) :
↑() m * ↑() m = 0

As well as being linear, ι m squares to zero.

theorem ExteriorAlgebra.comp_ι_sq_zero {R : Type u1} [] {M : Type u2} [] [Module R M] {A : Type u_1} [] [Algebra R A] (g : →ₐ[R] A) (m : M) :
g (↑() m) * g (↑() m) = 0
@[simp]
theorem ExteriorAlgebra.lift_symm_apply (R : Type u1) [] {M : Type u2} [] [Module R M] {A : Type u_1} [] [Algebra R A] :
∀ (a : →ₐ[R] A), ().symm a = { val := ↑(().symm a), property := (_ : ∀ (m : M), ↑(().symm a) m * ↑(().symm a) m = 0) }
def ExteriorAlgebra.lift (R : Type u1) [] {M : Type u2} [] [Module R M] {A : Type u_1} [] [Algebra R A] :
{ f // ∀ (m : M), f m * f m = 0 } ( →ₐ[R] A)

Given a linear map f : M →ₗ[R] A into an R-algebra A, which satisfies the condition: cond : ∀ m : M, f m * f m = 0, this is the canonical lift of f to a morphism of R-algebras from ExteriorAlgebra R M to A.

Instances For
@[simp]
theorem ExteriorAlgebra.ι_comp_lift (R : Type u1) [] {M : Type u2} [] [Module R M] {A : Type u_1} [] [Algebra R A] (f : M →ₗ[R] A) (cond : ∀ (m : M), f m * f m = 0) :
LinearMap.comp (AlgHom.toLinearMap (↑() { val := f, property := cond })) () = f
@[simp]
theorem ExteriorAlgebra.lift_ι_apply (R : Type u1) [] {M : Type u2} [] [Module R M] {A : Type u_1} [] [Algebra R A] (f : M →ₗ[R] A) (cond : ∀ (m : M), f m * f m = 0) (x : M) :
↑(↑() { val := f, property := cond }) (↑() x) = f x
@[simp]
theorem ExteriorAlgebra.lift_unique (R : Type u1) [] {M : Type u2} [] [Module R M] {A : Type u_1} [] [Algebra R A] (f : M →ₗ[R] A) (cond : ∀ (m : M), f m * f m = 0) (g : →ₐ[R] A) :
g = ↑() { val := f, property := cond }
@[simp]
theorem ExteriorAlgebra.lift_comp_ι {R : Type u1} [] {M : Type u2} [] [Module R M] {A : Type u_1} [] [Algebra R A] (g : →ₐ[R] A) :
↑() { val := , property := (_ : ∀ (m : M), g (↑() m) * g (↑() m) = 0) } = g
theorem ExteriorAlgebra.hom_ext {R : Type u1} [] {M : Type u2} [] [Module R M] {A : Type u_1} [] [Algebra R A] {f : →ₐ[R] A} {g : →ₐ[R] A} (h : ) :
f = g

See note [partially-applied ext lemmas].

theorem ExteriorAlgebra.induction {R : Type u1} [] {M : Type u2} [] [Module R M] {C : Prop} (h_grade0 : (r : R) → C (↑(algebraMap R ()) r)) (h_grade1 : (x : M) → C (↑() x)) (h_mul : (a b : ) → C aC bC (a * b)) (h_add : (a b : ) → C aC bC (a + b)) (a : ) :
C a

If C holds for the algebraMap of r : R into ExteriorAlgebra R M, the ι of x : M, and is preserved under addition and muliplication, then it holds for all of ExteriorAlgebra R M.

def ExteriorAlgebra.algebraMapInv {R : Type u1} [] {M : Type u2} [] [Module R M] :

The left-inverse of algebraMap.

Instances For
theorem ExteriorAlgebra.algebraMap_leftInverse {R : Type u1} [] (M : Type u2) [] [Module R M] :
Function.LeftInverse ExteriorAlgebra.algebraMapInv ↑(algebraMap R ())
@[simp]
theorem ExteriorAlgebra.algebraMap_inj {R : Type u1} [] (M : Type u2) [] [Module R M] (x : R) (y : R) :
↑(algebraMap R ()) x = ↑(algebraMap R ()) y x = y
@[simp]
theorem ExteriorAlgebra.algebraMap_eq_zero_iff {R : Type u1} [] (M : Type u2) [] [Module R M] (x : R) :
↑(algebraMap R ()) x = 0 x = 0
@[simp]
theorem ExteriorAlgebra.algebraMap_eq_one_iff {R : Type u1} [] (M : Type u2) [] [Module R M] (x : R) :
↑(algebraMap R ()) x = 1 x = 1
theorem ExteriorAlgebra.isUnit_algebraMap {R : Type u1} [] (M : Type u2) [] [Module R M] (r : R) :
IsUnit (↑(algebraMap R ()) r)
@[simp]
theorem ExteriorAlgebra.invertibleAlgebraMapEquiv_symm_apply_invOf_toQuot {R : Type u1} [] (M : Type u2) [] [Module R M] (r : R) :
∀ (x : ), ((↑(algebraMap R ()) r)).toQuot = Quot.mk () (↑(algebraMap R ()) r)
@[simp]
theorem ExteriorAlgebra.invertibleAlgebraMapEquiv_apply_invOf {R : Type u1} [] (M : Type u2) [] [Module R M] (r : R) :
∀ (x : Invertible (↑(algebraMap R ()) r)), r = (ExteriorAlgebra.algebraMapInv (↑(algebraMap R ()) r))
def ExteriorAlgebra.invertibleAlgebraMapEquiv {R : Type u1} [] (M : Type u2) [] [Module R M] (r : R) :
Invertible (↑(algebraMap R ()) r)

Invertibility in the exterior algebra is the same as invertibility of the base ring.

Instances For
def ExteriorAlgebra.toTrivSqZeroExt {R : Type u1} [] {M : Type u2} [] [Module R M] [] [] :

The canonical map from ExteriorAlgebra R M into TrivSqZeroExt R M that sends ExteriorAlgebra.ι to TrivSqZeroExt.inr.

Instances For
@[simp]
theorem ExteriorAlgebra.toTrivSqZeroExt_ι {R : Type u1} [] {M : Type u2} [] [Module R M] [] [] (x : M) :
ExteriorAlgebra.toTrivSqZeroExt (↑() x) =
def ExteriorAlgebra.ιInv {R : Type u1} [] {M : Type u2} [] [Module R M] :

The left-inverse of ι.

As an implementation detail, we implement this using TrivSqZeroExt which has a suitable algebra structure.

Instances For
theorem ExteriorAlgebra.ι_leftInverse {R : Type u1} [] {M : Type u2} [] [Module R M] :
@[simp]
theorem ExteriorAlgebra.ι_inj (R : Type u1) [] {M : Type u2} [] [Module R M] (x : M) (y : M) :
↑() x = ↑() y x = y
@[simp]
theorem ExteriorAlgebra.ι_eq_zero_iff {R : Type u1} [] {M : Type u2} [] [Module R M] (x : M) :
↑() x = 0 x = 0
@[simp]
theorem ExteriorAlgebra.ι_eq_algebraMap_iff {R : Type u1} [] {M : Type u2} [] [Module R M] (x : M) (r : R) :
↑() x = ↑(algebraMap R ()) r x = 0 r = 0
@[simp]
theorem ExteriorAlgebra.ι_ne_one {R : Type u1} [] {M : Type u2} [] [Module R M] [] (x : M) :
↑() x 1
theorem ExteriorAlgebra.ι_range_disjoint_one {R : Type u1} [] {M : Type u2} [] [Module R M] :

The generators of the exterior algebra are disjoint from its scalars.

@[simp]
theorem ExteriorAlgebra.ι_add_mul_swap {R : Type u1} [] {M : Type u2} [] [Module R M] (x : M) (y : M) :
↑() x * ↑() y + ↑() y * ↑() x = 0
theorem ExteriorAlgebra.ι_mul_prod_list {R : Type u1} [] {M : Type u2} [] [Module R M] {n : } (f : Fin nM) (i : Fin n) :
↑() (f i) * List.prod (List.ofFn fun i => ↑() (f i)) = 0
def ExteriorAlgebra.ιMulti (R : Type u1) [] {M : Type u2} [] [Module R M] (n : ) :
AlternatingMap R M () (Fin n)

The product of n terms of the form ι R m is an alternating map.

This is a special case of MultilinearMap.mkPiAlgebraFin, and the exterior algebra version of TensorAlgebra.tprod.

Instances For
theorem ExteriorAlgebra.ιMulti_apply {R : Type u1} [] {M : Type u2} [] [Module R M] {n : } (v : Fin nM) :
↑() v = List.prod (List.ofFn fun i => ↑() (v i))
@[simp]
theorem ExteriorAlgebra.ιMulti_zero_apply {R : Type u1} [] {M : Type u2} [] [Module R M] (v : Fin 0M) :
↑() v = 1
@[simp]
theorem ExteriorAlgebra.ιMulti_succ_apply {R : Type u1} [] {M : Type u2} [] [Module R M] {n : } (v : Fin ()M) :
↑() v = ↑() (v 0) * ↑() ()
theorem ExteriorAlgebra.ιMulti_succ_curryLeft {R : Type u1} [] {M : Type u2} [] [Module R M] {n : } (m : M) :
↑() m = ↑() ()
def TensorAlgebra.toExterior {R : Type u1} [] {M : Type u2} [] [Module R M] :

The canonical image of the TensorAlgebra in the ExteriorAlgebra, which maps TensorAlgebra.ι R x to ExteriorAlgebra.ι R x.

Instances For
@[simp]
theorem TensorAlgebra.toExterior_ι {R : Type u1} [] {M : Type u2} [] [Module R M] (m : M) :
TensorAlgebra.toExterior (↑() m) = ↑() m